Decomposition methods for discrete problems

Descripción del proyecto

La descomposición hace que los problemas sean más digeribles en el ámbito de la teoría de grafos

La teoría de grafos ha surgido como un área muy importante de las matemáticas discretas. Utiliza un enfoque geométrico para estudiar los objetos, lo que contribuye a clasificar y representar los datos como una colección de aristas o caminos y vértices o nodos. Ha encontrado aplicaciones en numerosos campos, desde la representación de las interrelaciones entre objetos en plataformas de redes sociales hasta la optimización de la ruta más corta en sistemas GPS. El proyecto BOBR, financiado con fondos europeos, se centra en ampliar los límites de los conocimientos existentes relativos a los métodos de descomposición en la teoría de grafos mediante el desglose de problemas grandes en otros más pequeños y manejables, con especial atención en los algoritmos paramétricos y de aproximación.

Objetivo

The main goal of the project is to radically expand our understanding of decomposition methods for discrete problems, with a particular focus on the design of parameterized and approximation algorithms on graphs. We will concentrate on four topics where we see a potential for either establishing new directions, or reaching far beyond the current state of the art.

(Beyond) Sparsity: The field of Sparsity is a rapidly developing area of graph theory that studies abstract notions of uniform sparseness in graphs and provides a wealth of tools for algorithm design. While there are still many unknowns within this field, we would like to reach beyond sparse graphs by developing a theory of well-structured dense graphs, inspired by the advances in Sparsity.

Parameterized dynamic algorithms: The idea of parameterization has so far received little attention in the field of dynamic algorithms. Our goal is to establish solid foundations for the direction of parameterized dynamic algorithms by providing dynamic variants of basic decomposition tools used in parameterized complexity.

Parameterization and approximation on planar graphs: The areas of parameterized algorithms and of approximation schemes on planar graphs share a core set of decomposition techniques and benefit from extensive cross-inspiration. We will approach several intriguing questions in this area while focusing on the idea of parameterized approximation schemes, where parameterization and approximation is explicitly combined.

Forbidding induced subgraphs: Structural graph theory offers a wealth of tools for understanding structure in graph classes characterized by forbidding induced subgraphs. This structure, while elusive and difficult to exploit, often leads to surprising tractability results. Motivated by recent advances, we propose to focus on finding general-use techniques for designing subexponential-time, approximation, and parameterized algorithms in this setting.